Question: Simplify the following expression: $ t = 3 - \dfrac{z + 7}{-9z + 4} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-9z + 4}{-9z + 4}$ $ \dfrac{3}{1} \times \dfrac{-9z + 4}{-9z + 4} = \dfrac{-27z + 12}{-9z + 4} $ Therefore $ t = \dfrac{-27z + 12}{-9z + 4} - \dfrac{z + 7}{-9z + 4} $ Now the expressions have the same denominator we can simply subtract the numerators: $t = \dfrac{-27z + 12 - (z + 7) }{-9z + 4} $ Distribute the negative sign: $t = \dfrac{-27z + 12 - z - 7}{-9z + 4}$ $t = \dfrac{-28z + 5}{-9z + 4}$ Simplify the expression by dividing the numerator and denominator by -1: $t = \dfrac{28z - 5}{9z - 4}$